# Homotopy sphere

__: Concept in algebraic topology: n-manifold with the same homotopy and homology groups as the n-sphere__

**Short description**In algebraic topology, a branch of mathematics, a *homotopy sphere* is an *n*-manifold that is homotopy equivalent to the *n*-sphere. It thus has the same homotopy groups and the same homology groups as the *n*-sphere, and so every homotopy sphere is necessarily a homology sphere.^{[1]}

The topological generalized Poincaré conjecture is that any *n*-dimensional homotopy sphere is homeomorphic to the *n*-sphere; it was solved by Stephen Smale in dimensions five and higher, by Michael Freedman in dimension 4, and for dimension 3 (the original Poincaré conjecture) by Grigori Perelman in 2005.

The resolution of the smooth Poincaré conjecture in dimensions 5 and larger implies that homotopy spheres in those dimensions are precisely exotic spheres. It is still an open question ((As of February 2019)) whether or not there are non-trivial smooth homotopy spheres in dimension 4.

## References

- ↑ A., Kosinski, Antoni (1993).
*Differential manifolds*. Academic Press. ISBN 0-12-421850-4. OCLC 875287946. http://worldcat.org/oclc/875287946.

## See also

Original source: https://en.wikipedia.org/wiki/Homotopy sphere.
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